311edo: Difference between revisions

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311edo is [[consistent]] through the [[41-odd-limit]] and nearly distinctly consistent through the [[27-odd-limit]] except for [[25/24]][[~]][[26/25]], [[tempering out]] [[625/624]] ({{S|25}}), and is a [[zeta gap edo]] and a [[zeta peak integer edo]]. This is because all [[harmonic]]s up to the 42nd, and all composite harmonics up to the 80th, have no more than ±25% error. Prime 73 is also unusually accurate, more so than all smaller primes. As a result, all ratios among those harmonics are mapped consistently, with errors lower than 1.929{{c}}. This means 311edo is a ''serendipitously'' efficient temperament for approximating the [[harmonic series]] and the [[41-limit]] in general, consistently and ''simply'', given how much harmonic content it approximates/represents for its size.

The smallest edo that comes closest to having a higher consistency is [[~~1600edo~~]] ([[37-odd-limit]]). The next edo with a truly higher [[consistency limit]] is [[17461edo|17461]] ([[45-odd-limit]]), though one may prefer [[20567edo|20567]] ([[57-odd-limit]]).

The smallest edo that comes closest to having a higher consistency is [[388edo]] ([[37-odd-limit]]). The next edo with a truly higher [[consistency limit]] is [[17461edo|17461]] ([[45-odd-limit]]), though one may prefer [[20567edo|20567]] ([[57-odd-limit]]).

311edo is also the smallest edo that is [[purely consistent]] on all the first 32 harmonics (in this case, up to the 42nd). The next edo with less maximum relative error is [[16808edo|16808]]. The smallest edo purely consistent on the first 64 harmonics is [[3159811edo|3159811]].

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 edo is [[consistent]] through the [[41-odd-limit]] and nearly distinctly consistent through the [[27-odd-limit]] except for [[25/24]][[~]][[26/25]], [[tempering out]] [[625/624]] ({{S|25}}), and is a [[zeta gap edo]] and a [[zeta peak integer edo]]. This is because all [[harmonic]]s up to the 42nd, and all composite harmonics up to the 80th, have no more than ±25% error. Prime 73 is also unusually accurate, more so than all smaller primes. As a result, all ratios among those harmonics are mapped consistently, with errors lower than 1.929{{c}}. This means 311edo is a ''serendipitously'' efficient temperament for approximating the [[harmonic series]] and the [[41-limit]] in general, consistently and ''simply'', given how much harmonic content it approximates/represents for its size.
-The smallest edo that comes closest to having a higher consistency is [[1600edo]] ([[37-odd-limit]]). The next edo with a truly higher [[consistency limit]] is [[17461edo|17461]] ([[45-odd-limit]]), though one may prefer [[20567edo|20567]] ([[57-odd-limit]]).
+The smallest edo that comes closest to having a higher consistency is [[388edo]] ([[37-odd-limit]]). The next edo with a truly higher [[consistency limit]] is [[17461edo|17461]] ([[45-odd-limit]]), though one may prefer [[20567edo|20567]] ([[57-odd-limit]]).
 edo is also the smallest edo that is [[purely consistent]] on all the first 32 harmonics (in this case, up to the 42nd). The next edo with less maximum relative error is [[16808edo|16808]]. The smallest edo purely consistent on the first 64 harmonics is [[3159811edo|3159811]].